Standing in the shower preparing to dry off, I consider myself as at my most lucid condition. But as I dry myself with a big fluffy towel, I tend to move on to another place on the towel when my hands feel any moisture. Thus, although I believe that the dryness or the lack thereof of the side of the towel that my hands can feel is unrelated to the side that is drying off my body, I am usually thinking about something else so I still move the towel. I won’t even try to figure out the function for this, but it is clearly a fallible indicator. Hogarth reminded me of this when he noted our poorer performance when dealing with nonlinear relationships, “Human achievement is lower when there are nonlinearities in the ecology.” (What has Brunswik’s Lens Model Taught?). This reminds me of derivative financial instruments. My intuition cannot handle a straddled put (I think I made that up.) I can learn and feed it in, but give me 15 seconds to figure it out and my performance will be worse than chance. This is kind of a big deal if John von Neumann’s analogy is correct: that studying nonlinear relationships is similar to studying non elephants.

The difference between linear and nonlinear can be somewhat arbitrary in my mind. We can transform nonlinear functions to make them linear, but this probably does not help our intuition. Similarly, we can sometimes approximate a nonlinear function with a couple of linear functions. The bottom line seems to be that linear implies cause and effect with more or less input of one variable creating more or less output of another.

Michael Mauboussin in the *Success Equation *suggests that activities can be placed at points on what he calls the luck-skill continuum. Knowing this location can help make accurate statistical predictions. The key to statistical prediction is knowing the weight for the base rate and the specific case. When skill plays the prime role, you can rely on specific evidence. If luck is the prime determinant, use the base rate. Activities that are stable and linear tend to be on the skill side. You can become an expert if cause and effect are clear and consistent and stable and linear activities allow this. A linear system is like the interaction of balls on a billiards table. The stock market is a nonlinear system so there is no reliable way to train your intuition to predict prices.

Mauboussin notes that power functions are those where an exponent determines the slope of the line. These nonlinear functions explain a diverse range of socially driven phenomena including rank and number of book sales, rank and frequency of citations for scientific papers, and the rank and number of deaths in terrorism and war. Alas, cause and effect are not clear. We see very few large values and lots of small values, so the average has no meaning for our linear driven brains.

Mauboussin also brings out Taleb’s 2 by 2 matrix with the dreaded fourth black swan box where payoffs are complex and distributions have fat tails(Hogarth calls about the same thing in his own 2 by 2: the quadrant of irrelevant feedback and exacting consequences “wicked”). So it seems that not only do our intuitive brains perform poorly with nonlinearity, but also our analytical statistical tools.

But then I get more confused. Mauboussin suggests that we become experts by training our intuition by deliberate practice often with a coach or teacher, but that the activity needs to be stable and linear or our performance won’t reliably improve. Hogarth seems to largely agree. Nevertheless, intuition is often seen as nonlinear while analysis is linear. That probably comes from our intuitions seeming to produce answers from a single point, while analysis must plod along in a linear fashion to try to ascertain cause and effect. Hogarth clearly identifies the Brunswik lens model as the sum of linear functions. Parallel Constraint Satisfaction appears linear to me based on the equations, but some of the transformations are nonlinear, and Andreas Glockner definitively gives his answer to the question:

Is Decision Making Based on Linear or Nonlinear Information Integration?

In behavioral decision research as well as in neuroscience, there has been much debate as to

whether information integration in perception and decision making is based on a linear aggregation of evidence or on a nonlinear integration process. Gold and Shadlen (2007) report findings that support linear evidence accumulation models, which have been also proposed in behavioral decision research (Busemeyer and Townsend 1993; Ratcliff et al. 1999). In this chapter, I have summarized behavioral findings in support of the nonlinear PCS rule that are coherent with neuroscientific models of synchronization and binding (Singer 2003). Although these findings cannot be easily explained by linear models, the debate between approaches can be expected to continue. However, I would like to highlight two very general points. First, mathematically, linear models are partial models of nonlinear models. Thus, nonlinear models can usually account for all findings of linear models, but not the other way around. Second, it is not possible to differentiate between linear and nonlinear models in simple decision tasks (which are, for practical reasons, commonly used in neuroscience research on apes) because in such tasks, the predictions of both classes of models converge. A differentiation is only possible in complex decision tasks that allow for nonlinear effects.

It is interesting to wonder if the brain or large portions of it can be nonlinear….logical extreme– quantum computer..while apparently functioning much better in a linear ecology. Of course, a linear ecology is probably the simplest to perform well in, so we are best at the exception rather than the rule.

Karelaia, N., Hogarth, R.(2008). “Determinants of Linear Judgment: A Meta-Analysis of Lens Model Studies.” *Psychological Bulletin*. Vol. 134, No. 3, 404–426.

Hogarth, R.(2001) * Educating Intuition. *Chicago and London: University of Chicago Press.

Mauboussin, M.(2012). *The Success Equation Untangling Skill and Luck in Business, Sports, and Investing.* Boston: Harvard Business Review Press.

Glockner, A.(2008). “How Evolution Outwits Bounded Rationality The Efficient Interaction of Automatic and Deliberate Processes in Decision Making and Implications for Institutions.” Preprints of the Max Planck Institute for Research on Collective Goods Bonn 2008/8, p 22.